A. Multicarrier Transmission Systems
A simplified model of a multicarrier transmission system is shown in FIG. 1. Multicarrier transmission uses frequency division multiplexing (FDM) to divide the transmission system into a set of frequency-indexed subchannels that appear to be modulated and demodulated independently. If the bandwidth of each subchannel is limited, subchannel distortion can be modeled by a single attenuation and phase-shift. With careful allocation of bits and transmit power to each subchannel, it is well known that such a system is capable of performing at the highest theoretical limits and that no other system can exceed its performance.
Data rates are maximized through multicarrier transmission systems by varying the number of bits per symbol, and the proportion of the total transmitted power that are allotted to each subchannel in accordance with the channel distortion and narrowband noise present in each subchannel. The aggregate bit rate is maximized if these variables are chosen so that the bit error rates in each of the subchannels are equal.
B. DMT Modulation
In DMT modulation, serial input data at a rate of M*.function..sub.s bits per second are grouped into M bits at a block ("symbol") rate of .function..sub.s symbols per second. The M bits within each block are further subdivided, m.sub.n bits allotted to the carrier .function..sub.c,n, to modulate the N.sub.c carriers, which are spaced across the usable transmission spectrum. The modulated carriers are summed before transmission, and must be separated in the receiver before demodulation (See FIG. 2.).
In practice, the modulation and demodulation process is implemented with Fast Fourier Transform (FFT) algorithms. Modulation is achieved with the Inverses Fast Fourier Transform (IFFT), which generates N.sub.samp samples of a transmit signal for each block of M bits. In the receiver, the received signal is demodulated by each of the N.sub.c carriers, and the m.sub.n bits are recovered from each carrier. Again, the preferred method of demodulation is to sample the received signal, group the samples into blocks, and perform an FFT.
In such a transmission system, the symbol rate, and the carrier frequency separation are typically equal, and the receiver uses N.sub.samp samples to retrieve the data.
C. Intersymbol Interference and Noise
This idealized system encounters two types of problems in practice. The first type of problem encountered by multicarrier transmission systems is intersymbol interference (ISI) whereby the symbol decoded at the receiver will include interference from previously transmitted symbols. This type of interference is further aggravated by the high sidelobes in the sub-bands provided by the Fourier transform.
Prior art systems solve the intersymbol interference problems by including a guard band between consecutive symbols. This method, however, reduces the amount of information that can be transmitted.
The second type of problem is caused by colored noise which results from the pickup of other communication signals that impinge on the communication path. These signals enter the system at points in the communication path that are not sufficiently shielded. Providing perfect shielding in long communication paths is not practical.
In principle, a multicarrier transmission system can detect the presence of a high noise signal in one sub-band and merely avoid transmitting data in that sub-band. In practice, this solution does not function properly because of the characteristics of the sub-bands obtained using Fourier transforms. The Fourier transform provides sub-bands that are isolated by only 13 dB. Hence, the sub-bands have sidelobes that extend into the neighboring channels (FIG. 3). A large noise signal in one channel will spill over into several channels on each side of the channel in question and thereby degrade the overall performance of the communication system.
D. Equalization of DMT System to Eliminate ISI
In DMT modulation, a vector of QAM encoded data is the input to the IFFT block, and the output will be a sum of sinusoids with data dependent amplitudes and phases. As mentioned above, ISI causes a part of one symbol to be corrupted by the previous symbol. However, if at the beginning of every symbol, its periodic extension, "cyclic prefix" (CP), of length equal to the length of the channel impulse response were inserted as a guard-band, then the rest of that symbol would be free from ISI (FIG. 4). The longer the symbol duration the smaller the overhead due to CP is. Unfortunately, there are certain limitations on how long a symbol can be. DMT equalizers operate to keep the overhead as small as possible by reducing the effective impulse response of the channel to a duration that is shorter than that of the CP. Shortening the impulse response of a distorting communications channel allows the use of a short, efficient cyclic prefix which is appended to a multicarrier signal.
In the ADSL implementation of DMT, the overhead due to the CP is about 6.25%. The only constraint for the equalizer, therefore, is that the channel frequency response must be shorter than CP. Letting H(e.sup.jw) be the frequency response of the system between IFFT and FFT blocks, in the absence of noise: ##EQU1## where X(e.sup.jw) is a wideband periodic signal and Y(e.sup.jw) is the corresponding received signal. H(z) is modeled as a ratio of two FIR filters F(z) and T(z) of orders n.sub.f and n.sub.t respectively. If T(z) is used as the equalizer, then the overall impulse response is equal to F(z). Now it is easy to see that the only constraint in modeling H(z)=F(z)/T(z) is that F(z) must be shorter than the chosen CP. From Eq. 1.1: EQU F(e.sup.jw)X(e.sup.jw)=T(e.sup.jw)Y(e.sup.jw). (Eq. 1.2)
In general it is not possible to find two polynomials F(z) and T(z) such that Eq. 1.2 is satisfied exactly. Therefore, there will always be some residual ISI.
E. Conventional Equalizer Training Algorithm
Eq. 1.2 can be solved using IIR adaptive techniques as described by Chow and Cioffi in U.S. Pat. No. 5,285,474, "Method For Equalizing a Multicarrier Signal in a Multicarrier Communication System." The objective is to satisfy equation Eq. 1.2 in the frequency-domain and use the FFT and IFFT to ensure that T(z) and F(z) have the desired time-domain durations. Let T, F and H be vectors (bold letters denote vectors) of samples of T(e.sup.jw), F(e.sup.jw) and H(e.sup.jw)=Y(e.sup.jw)/X(e.sup.jw) respectively (i.e. T.sub.i =T(e.sup.2.pi.i/N) and so on). The iterative technique proceeds as follows:
1. For a given T, compute F=(TH), where brackets () indicate that multiplication is done component-wise. PA1 2. IFFT(F)=f will in general have longer duration than CP, so it has to be windowed. This process involves searching for the window position which captures maximum energy of f and zeroing out components outside the window. As a starting point, assume that the optimal window position is already known. PA1 3. After this windowing, (1.2) is not satisfied anymore. Therefore, the error is provided by: EQU E=(FX)-(TY) (Eq. 1.3) PA1 and a Least Mean Square (LMS) algorithm is used to update T as follows T'=T+.mu.(EY*). (* denotes complex conjugation) PA1 4. Now IFFT(T)=t is longer than desired and it has to be windowed. Its window can be kept fixed, and the first n.sub.t entries are kept and all others zeroed out. PA1 5. Go to step 1.
These steps are repeated until convergence. There are no published proofs of convergence, and some authors have hypothesized, based on observations, that there may be local minima and that the algorithm may not find the smallest one.
Since vectors X and Y are used to estimate the transfer function, they have to be free of noise. X is the transmit signal and therefore, contains no noise. A noise-free version of the received signal Y is obtained by averaging a certain number of the noisy received signals over time. As a result of this averaging, the noise component of the received signal is lost and the training process does not take it into account. The conventional equalizer, therefore, maximizes signal to ISI ratio (S/I) only. In the presence of a cross-talk (colored noise), this approach does not minimize total error (due to ISI and noise). In an ideal DMT system, where different frequency bins do not overlap in frequency, it does not matter whether channel noise is white or colored because the SNR in a particular bin is affected by the magnitude of the noise Power Spectral Density (PSD) at that frequency only. Unfortunately, practical implementations of DMT using FFT have rather poor frequency selectivity (first side-lobe is at about -13 dB). Consequently, SNR in any bin is a function of the overall noise PSD. One way to deal with this problem is to use a more sophisticated implementation of the DMT system (for example, a maximally decimated filter bank approach as described in ANSI, "Network and customer installation interfaces: Asymmetric digital subscriber line (ADSL) metallic interface," in draft American National Standard for Telecommunications, vol. T1E1.4/94-007R8, 1995). Another way is to train the equalizer so as to maximize the signal with respect to the total distortion which consists of both ISI and cross-talk.
It is well known that an equalizer trained by the time-domain LMS algorithm, the most commonly used training algorithm, minimizes the total MSE. (See D. Falconer and F. Magee, Adaptive channel memory truncation for maximum likelihood sequence estimation," Bell System Tech. J., vol. 52, No. 9, pp. 1541-1562, November 1973. and While it is possible to use the time-domain LMS algorithm in this case, extensive simulations have shown that it does not perform as well as the frequency-domain training algorithm described below.
The objective of the present invention is to provide a DMT equalizer training algorithm which will provide a unique MMSE equalizer solution that accounts for both ISI and noise.